In deterministic theories, the state of any isolated system at any moment of time is determined by its state in any of the precedent moments of time. Such a case takes place in quantum mechanics; once the [Hamiltonian]] of the system is given and the state of the system, say, at time $t\!=\!0$ is specified, this determines the state of the system at any moment of time. In probablilistic theories, the concept of causality is not trivial, but still allow some confirmation and refutation based on methods of the theory of probability and mathematical statistics .
In general, it is assumed that every event has a cause, that had happened in a precedent moment of time, and, in principle, the affect of the cause to the event can be revealed at the detailed consideration. Such an assumption can be considered as postulate of causality. In short form this can be expressed in a form "Each event has a cause". .
Causality in physics
Causality is fundamental physical concept.
Various kinds of the Kramers-Kronig relation between real and imaginary parts of the Fourier-representation of any kind of succeptibility or permeabilty or polarization follow from the causality. Violation of the Kramers-Kronig relation can be interpreted as contradiction to the principle of causality.
Causality in history
Causality can be applied also to the historical events. For example, in 2009, the members of the Russian political party edro had organized meetings dedicated to the explosion of the train Nevsky express, after five days since the explosion. According to the Law, the notification about these meetings were submitted at least 10 days before the meetings, id est, before the explosion. As the principle of causality prohibits the consequence to happen before the cause, this indicates, that the organizers of meeting knew about the crash before it. In such a way, the principle of causality excludes the concept that the crash of Nevsky express resulted from the just accident (poor state of the railway, lack of maintenance of wagons, etc.), and allow to characterize that crash as Nevsky Express bombing (2009), id est, as act of terror against population Russia and the visitors.
Similar consideration can be applied to the catastrophe Katyn-2: according to several publications, dispatchers at the airport did not know the moment of crash of the airplane (nor number of the aborted intents to land that took place just before the crash). As the dispatchers are supposed to follow the plane by the radars and to be in the radio communication with the pilots, the detection of crash 15 minutes after it happened is not possible. The versions about 4 attempts to land and about crash at 2010 April 10 at 10:56 of the local time were published by the Russian representative, and the later official concept of the catastrophe does not explain such communications; they seem to happen without a cause. In such a way, the principle of causality contradicts the official version of the Russian representatives, suggested in some weeks after the accident. In such a way, the principle of causality allows to qualify that crash as terroristic act, and the official report of the Russian officials as inconsistent. Yet, except the terroristic act, no other scientific concept or that crash is available.
Physics without causality
There are attempts to construct physics without principle of causality: The principle of cause and effect is deeply rooted in human experience, so much so that it is routinely and tacitly assumed throughout science, even by scientists working in areas where time symmetry is theoretically ingrained, as it is in both classical and quantum physics. Experiments are said to cause their results, not the other way around. In this informal paper, we argue that this assumption should be replaced with a more general notion of mutual influence -- bi-directional relations or constraints on joint values of two or more variables. . The lack of verifiability and refutability of the concept of violation of causality does not allow to qualify it as scientific.
Causality in religions
In principle, deviations from the causality are allowed at the religious consideration: it can be postulated that God had some wish, and for this wish, he broke the laws of physics in order to see, what happens as a result. In this case, the will and actions of God (who, by definition, cannot be investigated by scientific methods) is declared as a cause.
Practically, various religions use causality as a basic concept. 
Humor about the causality
Only a cat may get pregnant without a cause; any other event has a cause.
- http://suppes-corpus.stanford.edu/articles/mpm/106-1.pdf Patrick Suppers. A probabilistic theory of causality. Helsinki, 1970.
- http://www.metaresearch.org/cosmology/physicshasitsprinciples.asp Michael Kroyter. Physics Has Its Principles. The Causality Principle. (Derived from conferences held at Cesena, Italy 1999/09/20 and Sutton, Montreal 2002/10/07) Minor update 2008/03/30. 'Physics is about everything one can see, hear or think about in the whole world. Mathematics is about everything!' -
- Richard Shoup. Physics Without Causality - Theory and Evidence. (No date is specified; the latest cite suggested corresponds to year 2006; cited by the state for year 2012)
- http://www.al-islam.org/philosophy/7.htm Ayatullah Muhammad Baqir as-Sadr. Our Philosophy. Chapter Three: The Principle of Causality. Translation, Introduction and Notes by Shams C.Inati. (No date is specified; perhaps, year 2000)
http://www.isn-oldenburg.de/~hilf/lehre/QM/QM/node6.html Eberhard Hilf. The Causality Principle. 2000-02-10.
http://plato.stanford.edu/entries/kant-hume-causality/ Graciela De Pierris, Michael Friedman. Kant and Hume on Causality. First published Wed Jun 4, 2008